Is GAv. berekhdam a cognate of the place-name Bactria?

In "Makers of History and Civilization..." L.A. Waddell first proposed that the place-name variously known as Ba-ra-ȧ-se-ki or Ba-ra-aḫ-se-ki, Pa-ra-aḫ-šum ki, Pȧ-ra-aḫ-si-im ki in Sumerian inscriptions is equated with the place-name Persia I did a further investigation to see if I could find recent works that agree with him and I found at least one: In "Mesopotamien: Akkade-Zeit und Ur III-Zeit, Annäherungen 3" (pub 1999) Walther Sallaberger-Aage Westnholz makes the same proposition. He equates Parahshum (Pa-ra-aḫ-šum) with the modern day province of Fars. This led me to the question: could this place-name Parahshum (the earlier form being Ba-ra-aḫ-se) be akin to the place-name Bactria? Bactria is equated with the Young Avestan place-name Bakhdi and according to Haug and other old sources this form equates to the Gathic Avestan form berekhdham. I don't know whether it was actually used as a place-name, but it seems plausible that berekhdham could have meant "highland" and is akin to the younger Avestan form berezant "mountain" a form which is well-established to have been derived from the PIE * bhereg̑h-, bherg̑hos- “high, noble; barrow, mountain.” Can anyone confirm whether GAv. berekhdam (which I think is transliterated bərəxđąm) is a derivative of PIE * bhereg̑h-?

This is Haugs translation of Gatha 44.7: That I shall ask Thee, tell it me right, O Ahura! Who has prepared the Bactrian (berekhdha) home with its properties? Who fashioned, by a weaving motion, the excellent son out of the father? To become acquainted with these things, I approach Thee, O Mazda, beneficent spirit! Creator of all beings!